Lower bounds for the matrix chain ordering problem: Extended abstract

Abstract

We show for any n-matrix instance of the Matrix Chain Ordering Problem (MCOP) to have certain parenthesizations of depth Θ(n) as solutions requires the matrix dimensions that are input to be exponential in n. That is, to ensure the MCOP can have any parenthesization as a solution, we must allow very expensive inputs. This exponential input lower bound implies a worst case bit complexity lower bound of Ω(n 2). This lower bound is parameterized and, depending on the optimal product tree depth, it goes from Ω(n 2) down to Ω(n lgn). Also, this paper gives a very simple Ω(n lg n) time lower bound for the MCOP for a class of algorithms on a comparison model with unit cost comparisons. This lower bound, to the authors' knowledge, captures all known algorithms for solving the matrix chain ordering problem, but does not consider bit operations. Finally, a trade-off is given between the input complexity lower bound and the atomic comparison based lower bound.